Modelling of miscible liquids with the Korteweg stress

Ilya Kostin[1]; Martine Marion; Rozenn Texier-Picard; Vitaly A. Volpert

  • [1] Université de Saint-Etienne, Équipe d’Analyse Numérique, 23 rue Paul MICHELON, 42023 Saint-Etienne Cedex 02, France;

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

  • Volume: 37, Issue: 5, page 741-753
  • ISSN: 0764-583X

Abstract

top
When two miscible fluids, such as glycerol (glycerin) and water, are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients exist during some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress. We prove the global existence and uniqueness of the solution for the associated initial-boundary value problem in a two-dimensional bounded domain. We study the longtime behavior of the solution and show that it converges to the uniform composition distribution with zero velocity field. We also present numerical simulations of miscible drops and show how transient interfacial phenomena can change their shape.

How to cite

top

Kostin, Ilya, et al. "Modelling of miscible liquids with the Korteweg stress." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.5 (2003): 741-753. <http://eudml.org/doc/245627>.

@article{Kostin2003,
abstract = {When two miscible fluids, such as glycerol (glycerin) and water, are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients exist during some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress. We prove the global existence and uniqueness of the solution for the associated initial-boundary value problem in a two-dimensional bounded domain. We study the longtime behavior of the solution and show that it converges to the uniform composition distribution with zero velocity field. We also present numerical simulations of miscible drops and show how transient interfacial phenomena can change their shape.},
affiliation = {Université de Saint-Etienne, Équipe d’Analyse Numérique, 23 rue Paul MICHELON, 42023 Saint-Etienne Cedex 02, France;},
author = {Kostin, Ilya, Marion, Martine, Texier-Picard, Rozenn, Volpert, Vitaly A.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {miscible liquids; Korteweg stress; drops},
language = {eng},
number = {5},
pages = {741-753},
publisher = {EDP-Sciences},
title = {Modelling of miscible liquids with the Korteweg stress},
url = {http://eudml.org/doc/245627},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Kostin, Ilya
AU - Marion, Martine
AU - Texier-Picard, Rozenn
AU - Volpert, Vitaly A.
TI - Modelling of miscible liquids with the Korteweg stress
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 5
SP - 741
EP - 753
AB - When two miscible fluids, such as glycerol (glycerin) and water, are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients exist during some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress. We prove the global existence and uniqueness of the solution for the associated initial-boundary value problem in a two-dimensional bounded domain. We study the longtime behavior of the solution and show that it converges to the uniform composition distribution with zero velocity field. We also present numerical simulations of miscible drops and show how transient interfacial phenomena can change their shape.
LA - eng
KW - miscible liquids; Korteweg stress; drops
UR - http://eudml.org/doc/245627
ER -

References

top
  1. [1] D.M. Anderson, G.B. McFadden and A.A. Wheeler, Diffuse interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1998) 139–165. 
  2. [2] L.K. Antanovskii, Microscale theory of surface tension. Phys. Rev. E 54 (1996) 6285–6290. 
  3. [3] J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial Free Energy. J. Chem. Phys. 28 (1958) 258–267. 
  4. [4] D. Joseph and M. Renardy, Fundamentals of two-fluid dynamics, Vol. II. Springer, New York (1992). Zbl0784.76003
  5. [5] D.J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l’on tient compte des forces capillaires causées par des variations de densité considérables mais connues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité. Arch. Néerl. Sci. Exactes Nat. Ser. II 6 (1901) 1–24. Zbl32.0756.02JFM32.0756.02
  6. [6] O.A. Ladyzhenskaya, Mathematical theory of viscous incompressible flow. Gordon and Breach (1963). Zbl0121.42701MR155093
  7. [7] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villars, Paris (1969). Zbl0189.40603MR259693
  8. [8] J. Pojman, N. Bessonov, R. Texier, V. Volpert and H. Wilke, Numerical simulations of transient interfacial phenomena in miscible fluids, in Proceedings AIAA, Reno, USA (January 2002). 
  9. [9] J. Pojman, Y. Chekanov, J. Masere, V. Volpert, T. Dumont and H. Wilke, Effective interfacial tension induced convection in miscible fluids, in Proceedings of the 39th AIAA Aerospace Science Meeting, Reno, USA (January 2001). 
  10. [10] P. Petitjeans, Une tension de surface pour les fluides miscibles. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 673–679. 
  11. [11] R. Temam, Navier-Stokes equations. Theory and numerical analysis. North-Holland Publishing Co., Amsterdam–New York, Stud. Math. Appl. 2 (1979). Zbl0426.35003
  12. [12] R. Temam, Navier-Stokes equations and nonlinear functional analysis. SIAM (1983). Zbl0833.35110
  13. [13] J.S. Rowlinson, Translation of J.D. van der Waals’ “The thermodynamic theory of capillarity under hypothesis of a continuous variation of density”. J. Statist. Phys. 20 (1979) 197. Zbl1245.82006
  14. [14] V. Volpert, J. Pojman and R. Texier-Picard, Convection induced by composition gradients in miscible liquids. C. R. Acad. Sci. Paris Sér. I Math. 330 (2002) 353–358. Zbl1076.76597

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.